Inapproximability of Congestion Games Alexander Skopalik, Berthold V - - PowerPoint PPT Presentation

Inapproximability of Congestion Games

Alexander Skopalik, Berthold V¨

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Department of Computer Science RWTH Aachen

Warwick 2007

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Network Congestion Games

Given a directed graph G = (V , E) with delay functions de : {1, . . . , n} → N, e ∈ E. Player i wants to allocate a path of minimal delay between a source si and a target ti.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Network Congestion Games

Given a directed graph G = (V , E) with delay functions de : {1, . . . , n} → N, e ∈ E. Player i wants to allocate a path of minimal delay between a source si and a target ti.

1,2,9 4,5,6 1,2,3 1,9,9 7,8,9

s t

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Network Congestion Games

Given a directed graph G = (V , E) with delay functions de : {1, . . . , n} → N, e ∈ E. Player i wants to allocate a path of minimal delay between a source si and a target ti.

1,2,9 4,5,6 1,2,3 1,9,9 7,8,9

s t

Game is called symmetric if all players have the same source/target pair.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Congestion Games - general Definition

Congestion game is a tuple G = (N, R, (Σi)i∈N , (dr)r∈R) with N = {1, . . . , n}, set of players R = {1, . . . , m}, set of resources Σi ⊆ 2[m], strategy space of player i dr : {1, . . . , n} → R, delay function or resource r For any state S = (S1, . . . , Sn) ∈ Σ1 × · · · Σn nr = number of players with r ∈ Si dr(nr) = delay of resource r

• r∈Si dr(nr) = delay of player i

S is Nash equilibrium if no player can unilaterally decrease its delay.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The transition graph

Definition The transition graph of a congestion game Γ contains a node for every state S and a directed edge (S, S ′) if S′ can be reached from S by the improvement step of a single player.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The transition graph

Definition The transition graph of a congestion game Γ contains a node for every state S and a directed edge (S, S ′) if S′ can be reached from S by the improvement step of a single player. The sinks of the transition graph are the Nash equilibria of G.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The transition graph

Definition The transition graph of a congestion game Γ contains a node for every state S and a directed edge (S, S ′) if S′ can be reached from S by the improvement step of a single player. The sinks of the transition graph are the Nash equilibria of G. Nash equilibria are local optima wrt Rosenthal’s potential function φ(S) =

• r∈R

nr(S)

• i=1

dr(i) .

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Complexity of computing equilibria

Known Results matroid games network games general games symmetric O(n2m2) polynomial asymmetric O(n2m2) matroid game results by [Ackermann, R¨

• glin, V. 2006]

all other results by [Fabrikant, Papadimitriou, Talwar 2004]

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Complexity of computing equilibria

Known Results matroid games network games general games symmetric O(n2m2) polynomial PLS-complete asymmetric O(n2m2) PLS-complete PLS-complete matroid game results by [Ackermann, R¨

• glin, V. 2006]

all other results by [Fabrikant, Papadimitriou, Talwar 2004]

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The complexity class PLS

PLS (Polynomial Local Search) PLS contains optimization problems with a specified neighborhood relationship Γ. It is required that there is a poly-time algorithm that, given any solution S, either computes a solution in Γ(S) with better objective value

• r certifies that S is a local optimum.

Examples: FLIP (circuit evaluation with Flip-neighborhood) Max-Sat with Flip-neighborhood Max-Cut with Flip-neighborhood TSP with 2-Opt-neighborhood Congestion games wrt improvement steps

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The complexity class PLS

PLS reductions Given two PLS problems Π1 and Π2 find a mapping from the in- stances of Π1 to the instances of Π2 such that the mapping can be computed in polynomial time, the local optima of Π1 are mapped to local optima of Π2, and given any local optimum of Π2, one can construct a local

• ptimum of Π1 in polynomial time.

Examples for PLS-complete problem: FLIP (via a master reduction) Max-Sat and POS-NAE-SAT Max-Cut

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Approximation of local search problems

Definition Consider any local search problem Π. Let α > 1. An α-approximation for an instance of Π is a state S with the property that every state in Γ(S) has a value of at most α times better than the value of S.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Approximation of local search problems

Definition Consider any local search problem Π. Let α > 1. An α-approximation for an instance of Π is a state S with the property that every state in Γ(S) has a value of at most α times better than the value of S. Orlin, Punnen, Abraham, Schulz 2004 There is a fully polynomial time approximation scheme for every problem in PLS.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Approximation of congestion games

Definition An α-approximate equilibrium, for α > 1, is a state S with the property that none of the players can improve its delay by a factor

• f more than α.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Approximation of congestion games

Definition An α-approximate equilibrium, for α > 1, is a state S with the property that none of the players can improve its delay by a factor

• f more than α.

Chien & Sinclair 2007 In any symmetric network congestion game in which all edges satisfy the β-bounded jump condition, i.e., de(i +1) ≤ βde(i) for all i ∈ N, there is a sequence of improvement steps converging in O(nβǫ−1 log(nD)) steps, where D is an upper bound on the maximum delay.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

New results

For any poly-time computable α > 1, finding an α-approximate Nash equilibrium in general congestion games with positive and increasing delay functions is PLS-hard.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

New results

For any poly-time computable α > 1, finding an α-approximate Nash equilibrium in general congestion games with positive and increasing delay functions is PLS-hard. For every n ∈ N, there is a congestion game with n players having a state with the property that every sequence of improvement steps leading from this state to an approximate equilibrium has exponential length in n.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

New results

For any poly-time computable α > 1, finding an α-approximate Nash equilibrium in general congestion games with positive and increasing delay functions is PLS-hard. For every n ∈ N, there is a congestion game with n players having a state with the property that every sequence of improvement steps leading from this state to an approximate equilibrium has exponential length in n. It is PSPACE-hard to compute an α-equilibrium reachable from a given state in a given congestion games.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Sketch of the analysis

We do a PLS-Reduction from FLIP. Definition (FLIP) An instance of the problem FLIP consists of a Boolean circuit C with input bits x1, . . . , xn and output bits y1, . . . , ym. The neighborhood N(x) of solution x is set of bit vectors x ′ that differ from x in one bit and c(x′) < c(x). We transform C into a congestion game G(C) such that Nash equilibria of G(C) correspond to a local optimum of fC.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Sketch of the analysis

We do a PLS-Reduction from FLIP. Definition (FLIP) An instance of the problem FLIP consists of a Boolean circuit C with input bits x1, . . . , xn and output bits y1, . . . , ym. The neighborhood N(x) of solution x is set of bit vectors x ′ that differ from x in one bit and c(x′) < c(x). We transform C into a congestion game G(C) such that Nash equilibria of G(C) correspond to a local optimum of fC. Delays of different strategies of any player in G(C) deviate at least by a factor of α. Thus all equilibria are α-approximate equilibria.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Representing circuits by congestion games

W.l.o.g. the circuit consists only of NAND-Gates. Sort the gates in reverse topological order. Design of i-th gate:

0/α 2i 0/α 2i 0/0/α 2i "zero strategies" "one strategies"

• utput player

input player a input player b

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Representing circuits by congestion games

W.l.o.g. the circuit consists only of NAND-Gates. Sort the gates in reverse topological order. Design of i-th gate:

0/α 2i 0/α 2i 0/0/α 2i "zero strategies" "one strategies"

• utput player

input player a input player b

Input players “trigger” the output player because of the reverse topological order of the gates.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The feedback problem

A simple Idea that does not work out ...

1 Construct a circuit S with n input and n output bits that, for

x ∈ {0, 1}n not being a local optimum, computes x ′ ∈ {0, 1}n with fC(x′) ≤ fC(x).

2 Represent S by a congestion game G(S). Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The feedback problem

A simple Idea that does not work out ...

1 Construct a circuit S with n input and n output bits that, for

x ∈ {0, 1}n not being a local optimum, computes x ′ ∈ {0, 1}n with fC(x′) ≤ fC(x).

2 Represent S by a congestion game G(S). 3 Additionally, ensure that the output bits in G(S) trigger the

input bits such that the circuit is in an equilibrium only when input and output bits are identical.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The feedback problem

A simple Idea that does not work out ...

1 Construct a circuit S with n input and n output bits that, for

x ∈ {0, 1}n not being a local optimum, computes x ′ ∈ {0, 1}n with fC(x′) ≤ fC(x).

2 Represent S by a congestion game G(S). 3 Additionally, ensure that the output bits in G(S) trigger the

input bits such that the circuit is in an equilibrium only when input and output bits are identical. If this construction would be possible then Nash equilibria correspond to local optimal.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

The feedback problem

A simple Idea that does not work out ...

1 Construct a circuit S with n input and n output bits that, for

x ∈ {0, 1}n not being a local optimum, computes x ′ ∈ {0, 1}n with fC(x′) ≤ fC(x).

2 Represent S by a congestion game G(S). 3 Additionally, ensure that the output bits in G(S) trigger the

input bits such that the circuit is in an equilibrium only when input and output bits are identical. If this construction would be possible then Nash equilibria correspond to local optimal. Problem: output players have much smaller delay differences than input players

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

PLS Reduction – High Level Description

Idea: Construct a congestion game simulating a processor The players n input players X1, . . . , Xn m clock players Z1, . . . , Zm (with large delay differences) gate players for several circuits the controller

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

PLS Reduction – High Level Description

Idea: Construct a congestion game simulating a processor The players n input players X1, . . . , Xn m clock players Z1, . . . , Zm (with large delay differences) gate players for several circuits the controller Two kinds of states Let M be a very large integer. In the expensive states at least one player has a delay of at least M. In the inexpensive states all players have a delay significantly smaller than M.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

PLS-Reduction – High Level Description

The expensive states do not contain Nash equilibria. So we only have to consider the inexpensive states.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

PLS-Reduction – High Level Description

The expensive states do not contain Nash equilibria. So we only have to consider the inexpensive states. In the inexpensive states ... the clock players Z1, . . . , Zm count downward while counting downward they trigger improvement steps (flips) of the input players X1, . . . , Xn

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

PLS-Reduction – High Level Description

The expensive states do not contain Nash equilibria. So we only have to consider the inexpensive states. In the inexpensive states ... the clock players Z1, . . . , Zm count downward while counting downward they trigger improvement steps (flips) of the input players X1, . . . , Xn the controller together with the circuits guarantee the property z ≥ fC(x) (upper bound condition) this ensures that the clock can only stop because it cannot trigger an improvement step

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

PLS-Reduction – High Level Description

The expensive states do not contain Nash equilibria. So we only have to consider the inexpensive states. In the inexpensive states ... the clock players Z1, . . . , Zm count downward while counting downward they trigger improvement steps (flips) of the input players X1, . . . , Xn the controller together with the circuits guarantee the property z ≥ fC(x) (upper bound condition) this ensures that the clock can only stop because it cannot trigger an improvement step consequently, every Nash equilibrium corresponds to a local

• ptimum of C

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Network Congestion Games

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Network Congestion Games

Conjecture For any poly-time computable α > 1, finding an α-approximate Nash equilibrium in network congestion games with positive and increasing delay functions is PLS-hard.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Network Congestion Games

Conjecture For any poly-time computable α > 1, finding an α-approximate Nash equilibrium in network congestion games with positive and increasing delay functions is PLS-hard. Theorem Let α, β > 1 be appropriate constants. For every n ∈ N, there is an n-player game with O(n) resources and β-jump bounded delay function such that there is a state that has distance exponential in n to all α-Nash equilibria.

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Open problems

Is there a set of reasonable assumptions on the delay functions such that

computing an approximate Nash equilibrium has polynomial complexity? improvement sequences reach an approximate Nash equilibrium after polynomially many steps?

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games

Open problems

Is there a set of reasonable assumptions on the delay functions such that

computing an approximate Nash equilibrium has polynomial complexity? improvement sequences reach an approximate Nash equilibrium after polynomially many steps?

What about other games? – e.g. the party affiliation game (Max-Cut)?

Alexander Skopalik, Berthold V¨

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Inapproximability of Congestion Games